Numerical stability of Euclidean algorithm over ultrametric fields
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 503-534.

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. ${ℤ}_{p}$ or $k\left[\phantom{\rule{-1.49994pt}{0ex}}\left[t\right]\phantom{\rule{-1.4444pt}{0ex}}\right]$ where $k$ is a field). We prove that Euclidean-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the distribution of the valuation of the subresultants of two random monic $p$-adic polynomials having the same degree.

Nous étudions le problème de la stabilité du calcul des résultants et sous-résultants des polynômes définis sur des anneaux de valuation discrète complets (e.g. ${ℤ}_{p}$ ou $k\left[\phantom{\rule{-1.49994pt}{0ex}}\left[t\right]\phantom{\rule{-1.4444pt}{0ex}}\right]$$k$ est un corps). Nous démontrons que les algorithmes de type Euclide sont très instables en moyenne et, dans de nombreux cas, nous expliquons comment les rendre stables sans dégrader la complexité. Chemin faisant, nous déterminons la loi de la valuation des sous-résultants de deux polynômes $p$-adiques aléatoires unitaires de même degré.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.989
Classification: 11S99, 11Y99, 68W30
Keywords: Euclidean algorithm, ultrametric precision, subresultants
Xavier Caruso 1

1 Université Rennes 1, IRMAR 35042 Rennes Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Xavier Caruso. Numerical stability of Euclidean algorithm  over ultrametric fields. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 503-534. doi : 10.5802/jtnb.989. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.989/

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